Fundamental Theorem of Calculus: Definite Intervals - AP Calculus BC
Card 1 of 30
What is the integral of $f(x) = 1$ from $a$ to $b$?
What is the integral of $f(x) = 1$ from $a$ to $b$?
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$b - a$. Antiderivative of 1 is $x$, difference of bounds gives length.
$b - a$. Antiderivative of 1 is $x$, difference of bounds gives length.
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Find the integral of $f(x) = 2x^2 - 3x + 1$ from 1 to 2.
Find the integral of $f(x) = 2x^2 - 3x + 1$ from 1 to 2.
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$\frac{7}{3}$. Evaluate $\frac{2x^3}{3} - \frac{3x^2}{2} + x$ at bounds.
$\frac{7}{3}$. Evaluate $\frac{2x^3}{3} - \frac{3x^2}{2} + x$ at bounds.
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What is the definite integral of $f(x) = 5$ from 0 to 3?
What is the definite integral of $f(x) = 5$ from 0 to 3?
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- Integral of constant function equals constant times interval length.
- Integral of constant function equals constant times interval length.
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Find the integral of $f(x) = 3x^2 + 2x$ from 0 to 2.
Find the integral of $f(x) = 3x^2 + 2x$ from 0 to 2.
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- Evaluate $x^3 + x^2$ at bounds 2 and 0.
- Evaluate $x^3 + x^2$ at bounds 2 and 0.
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What is the integral of $f(x) = e^{2x}$ from 0 to 1?
What is the integral of $f(x) = e^{2x}$ from 0 to 1?
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$\frac{e^2 - 1}{2}$. Antiderivative is $\frac{e^{2x}}{2}$, evaluate at bounds.
$\frac{e^2 - 1}{2}$. Antiderivative is $\frac{e^{2x}}{2}$, evaluate at bounds.
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What is the antiderivative of $f(x) = 2x^3$?
What is the antiderivative of $f(x) = 2x^3$?
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$\frac{x^4}{2} + C$. Power rule integration: increase exponent, divide by new exponent.
$\frac{x^4}{2} + C$. Power rule integration: increase exponent, divide by new exponent.
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What does the Fundamental Theorem of Calculus connect?
What does the Fundamental Theorem of Calculus connect?
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Differentiation and integration. Links the inverse operations of calculus.
Differentiation and integration. Links the inverse operations of calculus.
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Find the integral of $f(x) = 1 + x$ from 0 to 2.
Find the integral of $f(x) = 1 + x$ from 0 to 2.
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- Antiderivative is $x + \frac{x^2}{2}$, evaluate at bounds.
- Antiderivative is $x + \frac{x^2}{2}$, evaluate at bounds.
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Calculate the integral of $f(x) = x^{-1}$ from 1 to 4.
Calculate the integral of $f(x) = x^{-1}$ from 1 to 4.
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$\text{ln}(4)$. Same as $\int \frac{1}{x} dx$, antiderivative is $\ln(x)$.
$\text{ln}(4)$. Same as $\int \frac{1}{x} dx$, antiderivative is $\ln(x)$.
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Determine the integral of $f(x) = e^x$ from 0 to 1.
Determine the integral of $f(x) = e^x$ from 0 to 1.
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$e - 1$. Antiderivative of $e^x$ is $e^x$, then $e - 1$.
$e - 1$. Antiderivative of $e^x$ is $e^x$, then $e - 1$.
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Evaluate the integral of $f(x) = \frac{1}{x}$ from 1 to 2.
Evaluate the integral of $f(x) = \frac{1}{x}$ from 1 to 2.
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$\text{ln}(2)$. Antiderivative of $\frac{1}{x}$ is $\ln(x)$, then $\ln(2) - \ln(1)$.
$\text{ln}(2)$. Antiderivative of $\frac{1}{x}$ is $\ln(x)$, then $\ln(2) - \ln(1)$.
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Find the integral of $f(x) = 4x^3 - x^2 + 2$ from 0 to 1.
Find the integral of $f(x) = 4x^3 - x^2 + 2$ from 0 to 1.
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$\frac{13}{4}$. Evaluate $x^4 - \frac{x^3}{3} + 2x$ at bounds 1 and 0.
$\frac{13}{4}$. Evaluate $x^4 - \frac{x^3}{3} + 2x$ at bounds 1 and 0.
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Calculate the integral of $f(x) = \frac{1}{x^2}$ from 1 to 2.
Calculate the integral of $f(x) = \frac{1}{x^2}$ from 1 to 2.
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$\frac{1}{2}$. Antiderivative is $-\frac{1}{x}$, then $-\frac{1}{2} - (-1) = \frac{1}{2}$.
$\frac{1}{2}$. Antiderivative is $-\frac{1}{x}$, then $-\frac{1}{2} - (-1) = \frac{1}{2}$.
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Evaluate the integral of $f(x) = x^3 - 3x^2 + 2x$ from 0 to 1.
Evaluate the integral of $f(x) = x^3 - 3x^2 + 2x$ from 0 to 1.
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- Evaluate $\frac{x^4}{4} - x^3 + x^2$ at bounds 1 and 0.
- Evaluate $\frac{x^4}{4} - x^3 + x^2$ at bounds 1 and 0.
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Determine the integral of $f(x) = x^4$ from 0 to 1.
Determine the integral of $f(x) = x^4$ from 0 to 1.
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$\frac{1}{5}$. Antiderivative is $\frac{x^5}{5}$, then $\frac{1}{5} - 0$.
$\frac{1}{5}$. Antiderivative is $\frac{x^5}{5}$, then $\frac{1}{5} - 0$.
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What is the integral of $f(x) = 4x$ from 1 to 3?
What is the integral of $f(x) = 4x$ from 1 to 3?
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- Antiderivative is $2x^2$, then $18 - 2 = 16$.
- Antiderivative is $2x^2$, then $18 - 2 = 16$.
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Find the value of the integral of $f(x) = x^2 - x$ from 0 to 3.
Find the value of the integral of $f(x) = x^2 - x$ from 0 to 3.
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$\frac{9}{2}$. Evaluate $\frac{x^3}{3} - \frac{x^2}{2}$ at bounds 3 and 0.
$\frac{9}{2}$. Evaluate $\frac{x^3}{3} - \frac{x^2}{2}$ at bounds 3 and 0.
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Calculate the integral of $f(x) = \frac{1}{x}$ from 1 to $e$.
Calculate the integral of $f(x) = \frac{1}{x}$ from 1 to $e$.
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- Antiderivative of $\frac{1}{x}$ is $\ln(x)$, then $\ln(e) - \ln(1) = 1$.
- Antiderivative of $\frac{1}{x}$ is $\ln(x)$, then $\ln(e) - \ln(1) = 1$.
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Identify the antiderivative given $f(x) = 2x$.
Identify the antiderivative given $f(x) = 2x$.
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$F(x) = x^2 + C$. The antiderivative of $2x$ is $x^2$ plus a constant.
$F(x) = x^2 + C$. The antiderivative of $2x$ is $x^2$ plus a constant.
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Evaluate the integral of $f(x) = x^3$ from 0 to 2.
Evaluate the integral of $f(x) = x^3$ from 0 to 2.
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- Antiderivative is $\frac{x^4}{4}$, then $\frac{16}{4} - 0 = 4$.
- Antiderivative is $\frac{x^4}{4}$, then $\frac{16}{4} - 0 = 4$.
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What is the antiderivative of $f(x) = 2x^3$?
What is the antiderivative of $f(x) = 2x^3$?
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$\frac{x^4}{2} + C$. Power rule integration: increase exponent, divide by new exponent.
$\frac{x^4}{2} + C$. Power rule integration: increase exponent, divide by new exponent.
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What does the Fundamental Theorem of Calculus connect?
What does the Fundamental Theorem of Calculus connect?
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Differentiation and integration. Links the inverse operations of calculus.
Differentiation and integration. Links the inverse operations of calculus.
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What is the integral of $f(x) = 1$ from $a$ to $b$?
What is the integral of $f(x) = 1$ from $a$ to $b$?
Tap to reveal answer
$b - a$. Antiderivative of 1 is $x$, difference of bounds gives length.
$b - a$. Antiderivative of 1 is $x$, difference of bounds gives length.
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Determine the integral of $f(x) = e^x$ from 0 to 1.
Determine the integral of $f(x) = e^x$ from 0 to 1.
Tap to reveal answer
$e - 1$. Antiderivative of $e^x$ is $e^x$, then $e - 1$.
$e - 1$. Antiderivative of $e^x$ is $e^x$, then $e - 1$.
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Evaluate the integral of $f(x) = \frac{1}{x}$ from 1 to 2.
Evaluate the integral of $f(x) = \frac{1}{x}$ from 1 to 2.
Tap to reveal answer
$\text{ln}(2)$. Antiderivative of $\frac{1}{x}$ is $\ln(x)$, then $\ln(2) - \ln(1)$.
$\text{ln}(2)$. Antiderivative of $\frac{1}{x}$ is $\ln(x)$, then $\ln(2) - \ln(1)$.
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Find the integral of $f(x) = 4x^3 - x^2 + 2$ from 0 to 1.
Find the integral of $f(x) = 4x^3 - x^2 + 2$ from 0 to 1.
Tap to reveal answer
$\frac{13}{4}$. Evaluate $x^4 - \frac{x^3}{3} + 2x$ at bounds 1 and 0.
$\frac{13}{4}$. Evaluate $x^4 - \frac{x^3}{3} + 2x$ at bounds 1 and 0.
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Identify the antiderivative given $f(x) = 2x$.
Identify the antiderivative given $f(x) = 2x$.
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$F(x) = x^2 + C$. The antiderivative of $2x$ is $x^2$ plus a constant.
$F(x) = x^2 + C$. The antiderivative of $2x$ is $x^2$ plus a constant.
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Calculate the integral of $f(x) = \frac{1}{x}$ from 1 to $e$.
Calculate the integral of $f(x) = \frac{1}{x}$ from 1 to $e$.
Tap to reveal answer
- Antiderivative of $\frac{1}{x}$ is $\ln(x)$, then $\ln(e) - \ln(1) = 1$.
- Antiderivative of $\frac{1}{x}$ is $\ln(x)$, then $\ln(e) - \ln(1) = 1$.
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Find the value of the integral of $f(x) = x^2 - x$ from 0 to 3.
Find the value of the integral of $f(x) = x^2 - x$ from 0 to 3.
Tap to reveal answer
$\frac{9}{2}$. Evaluate $\frac{x^3}{3} - \frac{x^2}{2}$ at bounds 3 and 0.
$\frac{9}{2}$. Evaluate $\frac{x^3}{3} - \frac{x^2}{2}$ at bounds 3 and 0.
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What is the integral of $f(x) = 4x$ from 1 to 3?
What is the integral of $f(x) = 4x$ from 1 to 3?
Tap to reveal answer
- Antiderivative is $2x^2$, then $18 - 2 = 16$.
- Antiderivative is $2x^2$, then $18 - 2 = 16$.
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